1. Introduction
An obvious approach to achieving independent core propagation in a multicore fiber is to engineer a long coupling length by increasing the core separation and the mode confinement. This method, however, places restrictions on the fiber design as well as how densely the cores can be packed in a fiber. An alternative solution is to decrease the efficiency of the coupling such that an insignificant amount of power is transferred between the cores regardless of the coupling length or the rate of the coupling. Our results show that structural nonuniformities in twocore PCFs that could be a consequence of normal fabrication processes have the potential to drastically reduce the efficiency of the coupling allowing coupling to be practically ignored. The imperfections in the cladding cause the two cores to become decoupled and, as a result, light propagates essentially independently in each core.
2. Method
In order to understand how the introduction of nonuniformities affects the core coupling in a twocore PCF, we use a vector normal mode expansion to compare how power in each core fluctuates as a function of the propagation distance for many randomly generated fibers with imperfect lattice structures and for fibers without imperfections.
The PCFs simulated are composed, in crosssection, of air holes arranged in a triangular lattice around two silica defects separated by a single air hole, as shown in
Figs. 1(a) and
1(b). The x and y axes are also defined in 1(a). The guided field in the waveguide,
E⃗(
x,
y,
z) and
H⃗(
x,
y,
z), is expressed as an expansion of the bound modes of a particular twocore PCF following Eq. (
1) [
8
8
.
A.
Snyder
and
J.
Love
,
Optical Waveguide Theory
(
Kluwer, London
,
1983
).
,
9
9
.
C.
Vassallo
,
Optical Waveguide Concepts
(
Elsevier, Amsterdam
,
1991
).
]:
Fig. 1. The xpolarized modes (E_{x} is much larger than E_{y}) of the fundamental mode group for a perfect lattice twocore PCF with d/Λ=0.90 and Λ=2.5μm. (a) and (b) display the PCF crosssection and the real xcomponenet of the electric field; the white circles are air holes. (c) and (d) show the profile of the field along y=0; (a) and (c) are a symmetric mode, while (b) and (d) are antisymmetric.
The power as a function of z in a particular core is calculated from the Poynting vector using Eq. (
4):
3. Results
Fig. 2. The normalized power in each core is plotted versus the propagation distance. The coupling efficiency changes from (a) 1 to (b) 0.20 when a random variation of 2.2% in the air hole size is introduced, Λ = 2.5 μm and d/Λ = 0.58.
Fig. 3. The average coupling efficiency (a) and the average coupling length, normalized to the value for a perfectly uniform structure, (b) are plotted vs. d/Λ for a percentage variation of 1% (solid line) and 4% (dashed line), Λ = 2.5 μm. The lines have been added to facilitate the eye.
The coupling length exhibits a similar response to lattice variations, as shown in
Fig. 3(b), where the average normalized coupling length is plotted versus d/Λ. In order to examine the relationship between these two properties of twocore PCFs, the coupling efficiency is plotted versus the coupling length in
Fig. 4 for different values of d/Λ. Each marker represents the coupling efficiency and the coupling length of a randomly generated twocore fiber with variations of 0.67% to ~4% in either the air hole size or the air hole separation. Fibers with large air holes (bottom right) typically exhibit a coupling efficiency of less than 1%.
All the structures examined in
Fig. 4 are multimoded. By increasing the wavelength to pitch ratio (i.e. the normalized wavelength) for the relative air hole sizes studied here, or by decreasing the value of d/Λ into the endlessly single mode regime, a single mode will be guided. Changing the structure by increasing the normalized wavelength results in a larger mode size as does decreasing the air hole size; therefore, altering the normalized wavelength in this manner should follow the same trend shown in
Fig. 4 for changes in d/Λ. This is indeed the case. Data in
Fig. 4 moves up and to the left as the structure becomes less multimoded because of changes in the air hole size or in the normalized wavelength. This behavior demonstrates that twocore fibers become less sensitive to variations as the coupling length decreases and the mode field area becomes larger.
Upon inspection of the plots in
Fig. 4, the efficiency is noticed to decrease at a quicker rate than the coupling length in a clear power relationship, inferred from the linear distribution of data points in the loglog plots. The fact that the data exhibits similar slopes across all plots indicates that the nature of the relationship between these parameters does not appear to depend on the relative air hole size. A linear fit of the loglog data reveals an approximately quadratic relationship between the two parameters, as evidenced by the slope values given in
Table 1 shown for each of the plots in
Fig. 4. In order to develop an understanding of this relationship, we applied coupled mode theory to twocore PCFs with imperfect lattice structures.
Fig. 4. The coupling efficiency vs. the x polarization coupling length for Λ = 2.5 μm and d/Λ = 0.58, 0.70, 0.75, 0.80, 0.85 and 0.90 from upper left to lower right. Note that the axis scaling differs for each plot.
Table 1. The slopes of linear fits to the loglog data of Fig. 4. 
 
4. Couple mode theory
In coupled mode theory (CMT), each core is treated as an independent waveguide that is perturbed by the presence of fields propagating in the other core. When the field from one core enters the high index core region of the second waveguide, it becomes a source for a new field. The total field solution of a system of two waveguides, label them
a and
b, is approximated as a linear combination of the mode fields of the individual waveguides [
11
11
.
S. L.
Chuang
,
Physics of Optoelectronic Devices
(
WileyInterscience
,
1995
).
], as shown in Eq. (
5).
In the weak coupling regime, the cross power (proportionally to the area integrals of
e→ta
(
x,
y)×
h→tb
(
x,
y) and
e→tb
(
x,
y)×
h→ta
(
x,
y)) is ignored and the amplitudes of the two cores
obey the coupled amplitude equations of Eq. (
6) as derived from the general reciprocity relation [
11
11
.
S. L.
Chuang
,
Physics of Optoelectronic Devices
(
WileyInterscience
,
1995
).
].
where
β_{a}
and
β_{b}
are the propagation constants for the individual, independent waveguides and the coupling coefficients, K
_{ba} and K
_{ab}, are proportional to the overlap integral of the mode fields of the individual waveguides in each core [
10
10
.
A.
Barybin
and
V.
Dmitriev
,
Modern Electrodynamics and CoupledMode Theory: Application to GuidedWave Optics
(
Rinton Press
,
2002
).
,
11
11
.
S. L.
Chuang
,
Physics of Optoelectronic Devices
(
WileyInterscience
,
1995
).
]. Solving this system of equations and applying the initial condition that light is incident on waveguide
a at z = 0, or a(0) = 1 and b(0) = 0, produces the expression for the power in waveguide
b given in Eq. (
7) [
11
11
.
S. L.
Chuang
,
Physics of Optoelectronic Devices
(
WileyInterscience
,
1995
).
]:
The coupling properties of the twocore PCF are again defined from the power transfer, P
_{b}(z). The coupling length is the distance, L
_{c}, at which the power in waveguide
b has oscillated to its first maximum, ie.
ψ/L
_{c} =
π/2. The coupling efficiency can be defined from Eq. (
7) as the maximum possible value for the power in waveguide
b, or K
_{ab}/
ψ
^{2}. As the mismatch between the modes,
β_{b}

β_{a}
, increases, it can be seen from these relationships that the coupling efficiency and the coupling length will decrease. When the variable ψ is solved for in the expression for the coupling length and substituted into the expression for the coupling efficiency, the efficiency is shown to be proportional to the square of the coupling length:
Improved coupled mode theory, where the coupling is no longer assumed to be weak, results in the same relationship between the efficiency and the coupling length as in Eq. (
8), but with a different proportionality constant still dependent on the mode overlap that is modified by an amount determined by the cross power [
10
10
.
A.
Barybin
and
V.
Dmitriev
,
Modern Electrodynamics and CoupledMode Theory: Application to GuidedWave Optics
(
Rinton Press
,
2002
).
,
11
11
.
S. L.
Chuang
,
Physics of Optoelectronic Devices
(
WileyInterscience
,
1995
).
]. For simplicity, we will continue without including the effects of the cross power. The quadratic relationship of Eq. (
8) substantiates the relationship predicted from the power law fits of the plots in
Fig. 4 and from the values in
Table 1.
The applicability of CMT can be further assessed by comparing the coupling length, L
_{c}, as calculated from CMT using Eq. (
8) to the value derived directly through the multipole method for a perfect structure. Beginning with Eq. (
8), if K
_{ab} is assumed to be constant, a quadratic polynomial fit, where the linear and zeroth order terms in the polynomial are forced to be zero, can be applied to the data of
Fig. 4 and the fit parameter can be used to estimate K
_{ab}. The assumption that K
_{ab} is constant claims that the induced structural perturbations do not significantly change the amount of overlap between the mode fields of the two cores from that of a uniform twocore PCF.
Table 2 lists the fit parameters from the quadratic fit for each ratio of d/Λ studied and the coupling coefficient, K
_{ab}, as solved for from these values. An estimate of the coupling length for a perfect fiber can be calculated from K
_{ab} using CMT and Eq. (
7) with a mismatch set to zero. These values are compared in the final two columns of
Table 2 with the coupling length as determined directly from twocore fibers with a perfectly uniform photonic crystal lattice. This comparison implies that CMT in the weak coupling regime adequately describes the behavior of these twocore PCFs when nonuniformities are present. The discrepancy between the values for L
_{c} from the two methods most likely results from neglecting the cross power and from the parameter K
_{ab} not remaining constant for all imperfect structures. In fact, for a few cases, generated imperfect twocore PCFs exhibited longer coupling lengths than a fiber with a perfectly uniform lattice; in order to satisfy Eq. (
8) without predicting an efficiency greater than one, K
_{ab} would have to be much different than that of the perfect fiber. The data from these structures was left out of the fit on the basis that it did not satisfy the condition that K
_{ab} be approximately constant. Revisiting the power law fit results shown in
Table 1 with these data points removed leads to fits that are closer to quadratic; for example, for d/Λ=0.58 and d/Λ=0.70, the slopes increase to 1.98 and 1.99 respectively.
Table 2. The fit parameter, or the coefficient of a quadratic fit to the data in Fig. 4, is listed according to the relative air hole size. Kab is calculated from the fit parameter according to Eq. (8). The final columns compare the coupling length as calculated from the fit using CMT with the value determined directly from the perfect structure. 
 
5. Discussion
Nonuniformities in the lattice structure of twocore PCFs cause the cores to be no longer identical, thus creating a mismatch in the indices of the fundamental modes of the two individual cores. A nonzero mismatch results in a decreased coupling length and coupling efficiency as compared with the values that would be predicted from a perfect structure with no mismatch. The coupling properties of a twocore fiber structure with imperfections are determined by the parameter ψ, as defined in Eq. (
7), which depends on both the mismatch and K
_{ab}. Thus, knowledge of K
_{ab} alone, which can be gained from simulations based on a perfect twocore fiber, is not sufficient to accurately predict the coupling behavior of a particular PCF that is not completely uniform. It is the magnitude of the mismatch relative to K
_{ab} that determines how different the core coupling will be from that of a perfect structure. If K
_{ab} is small relative to the induced mismatch, the fiber is more sensitive and the coupling properties will change significantly when nonuniformities—and therefore core mismatch—are introduced. For example, K
_{ab}, which is proportional to the mode overlap, will be smaller when the modes are well confined, which is generally true of PCFs with large air holes (see
Table 2). The magnitude of the mismatch, or β
_{b}β
_{a}, can be estimated from the birefringence induced in a single core PCF when nonuniformities are introduced into the cladding structure. From Ref. [
4],
Fig. 2, the average induced birefringence of PCFs with d/Λ = 0.70 and d/Λ = 0.90 for Λ=2.5μm are shown to be of approximately the same order of magnitude, ∆n~1e4, across percentage variations of 0.67% to 4%. It can be seen in
Table 2 that this ∆n is two orders of magnitude larger than the coupling coefficient, K
_{ab}, for d/Λ = 0.90, while it is actually slightly smaller than K
_{ab} for d/Λ = 0.70. Therefore, the same mode mismatch between two cores will result in a much greater change in the coupling properties when the modes are tightly confined (d/Λ = 0.90). Clearly, small perturbations have a greater impact on fibers with larger air holes because they are characterized by a relatively large mismatch and a small overlap.
Fig. 5. K_{ab}, as computed from a perfect structure, (dashed lines) and ∆β, calculated from the birefringence induced from variations in the lattice of a single core PCF, (solid lines) are plotted versus the normalized wavelength. ∆β is the average value for 20 structures with variations of 1% in the air hole separation.
Consequences of the nonidentical nature of the cores are also manifested through a modification of the mode structure. Exploring the change in the mode shape provides a more intuitive understanding of the impact of nonuniformities on the coupling behavior of twocore PCFs. As stated previously, the eigenmodes of the unperturbed twocore system are perfectly symmetric or antisymmetric in amplitude and distribute energy evenly between the two cores. However, imperfections in the lattice cause the eigenmodes to evolve towards those of the decoupled cores as evidenced by the zcomponent of the Poynting vector shown in
Fig. 6 for a fiber with d/Λ=0.90 and variations of 0.67%. The energy of each mode is almost completely concentrated in one core.
Figure 7 shows the extinction ratio between the cores, or the ratio of energy in one core to that in the other, for fibers with different relative air hole sizes and for two degrees of variation in the fiber structure. Each marker is an average over thirty structures. As the percentage of variation in the lattice structure increases, and as the ratio of d/Λ increases, the mode energy becomes more concentrated in one core or the other. When d/Λ is greater than 0.80, the amount of energy localized in one core is, on average, at least three orders of magnitude greater than the energy in the other core for all the fundamental modes of the fiber, even when variations are as small as 1%. The two modes in the fundamental mode group with the highest indices will be almost entirely right (or left) modes while the remaining two modes will be left (or right) modes. From a scalar point of view, this change in the eigenmode structure explains the observed decrease in the coupling efficiency since the singlecore input field closely resembles one of the eigenmodes of the perturbed structure. Light that is incident on one of the cores will now couple predominantly into an eigenmode of the system and travel along the fiber according to the propagation constant of this mode with an almost constant transverse energy profile. Large air hole structures are more sensitive to decoupling because the modes are already well confined in the individual cores.
Fig. 6. A crosssectional view of the zcomponent of the real part of the Poynting vector for two of the modes from the highest index mode group of a twocore PCF with Λ = 2.5 m and d/Λ = 0.90 and nonuniformities of 0.67% appears in (a) and (b), while (c) and (d) show the profile along the line where y = 0. Similar behavior is observed for the two other modes in the group.
Fig. 7. The ratio of power in the left core to that in the right core for a mode that is predominantly left is shown as a function of the normalized air hole size, d/Λ, for a percentage variation of 1% (solid line) and 4% (dashed line), Λ = 2.5 μm. The lines have been added to facilitate the eye. The value for a perfect twocore fiber is 1.
For applications where independent core propagation is a requirement, two opposing approaches exist. A long coupling length achieved by increasing the core separation is the obvious method for obtaining this effect. The results presented in this paper, however, illuminate a more simple solution which is to reduce the coupling efficiency. The cores can be packed very closely but in a photonic crystal lattice that is very sensitive to perturbations, such as a PCF with a high ratio of d/Λ or well confined modes. When even slight nonuniformities are present, the two cores in these PCFs can be assumed to be decoupled and light will propagate independently in each core. Typical variations in the structure due to fabrication can actually drastically reduce the coupling length; however, since the efficiency of the coupling is approximately zero, all coupling can be essentially ignored.
We also note that due to the random nature in which the variations were imposed on the structures simulated in this paper, the results should remain robust to additional perturbations not accounted for, such as bending.